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The mathematics community at large seems pretty satisfied right now with the common practice of 1. starting with some axioms and 2. deriving theorems from them by employing some logic. All mathematics is done like this (as far as I know).

Here's the problem I see with this, specifically in the logic part (although I'm not so confident that I can explain it clearly). In defining a logic, you say things like, "A theory T is called a formal theory, if and only if there is an algorithm allowing to verify, is a given text a correct proof via principles of T, or not" (I just copied that from a webpage, so don't worry about the details). I'm focusing specifically on the "if and only if" part. What does "if and only if" mean, logically (i.e. how do I apply it)? Well, It means different things in different logics (our common notion of it changes especially in paraconsistent logics). So do words like "and," "or," and "is." For example, in our daily speaking language when we say "the light is on," we would consider that equivalent to "the light is not off." Not so if we were using a paraconsistent logic. So why assume, in important definitions such as the above, that we are using some agreed upon logic (the logic people normally assume is Aristotelian logic, which assumes the law of non-contradiction and the law of the excluded middle).

Here's my attempt to say it in one sentence with commentary: Using a logic (i.e. applying it to axioms to derive theorems) necessarily entails using another logic (not necessarily the same as the other one) that is not formally defined (it's not that you can't formally define it, but that you are not using it formally but casually, assuming that Aristotelian logic is the "right logic").

Thus, if I applied a logic to some axioms using a paraconsistent logic (which allows contradictions to a degree) instead of Aristotelian logic, a different theory would emerge.

I guess it seems to me that logic is so embedded in our natural language that it's impossible to define a truly "formal" logic in the sense that it does not assume anything (which is precisely what mathematicians have desired to do); even a formal logic assumes the casual use of another logic.

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Can you formulate a specific question? Your title might be considered subjective and argumentative as it is. See the FAQ. –  András Bátkai Oct 25 '12 at 8:23
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And the preface to the logic part of Nicolas Bourbaki's treaty (Ens, ch.I) says "we will not discuss here the possibility of teaching (or describing) mathematics to someone who couldn't read, write or count" –  Feldmann Denis Oct 25 '12 at 8:29
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In order to work in a fixed axiomatic theory, say ZFC, you don't need to reason about the theory, you just reason in the theory. You thus don’t need any of the mathematical logic machinery which treats theories as objects and studies their general properties. You only need to be able to recognize a valid proof, i.e., to be familiar with the axioms and derivation rules (“proof techniques”) of your theory. –  Emil Jeřábek Oct 25 '12 at 10:59
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"All mathematics is done like this (as far as I know)." Then perhaps you should find out a little more about how mathematics is done. –  Gerry Myerson Oct 25 '12 at 12:00
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I would like to keep this question, please, because it is a common worry by many people. I can answer this. I think it should be answered. –  Andrej Bauer Oct 25 '12 at 12:01
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1 Answer

No, they are not shaky.

I answered a similar question before, but let me try again.

You make a number of assumptions about logic. You are assuming that it is a goal of logic to bootstrap itself and mathematics starting from nothing. You think that a logician is not allowed to use the words "or" and "and" until he has somehow defined them without using them. You assume that "it is logics all the way down", that is to say, that the only way to define a logic is to use another logic. All these assumptions are incorrect.

You want logicans to impart on you a feeling of absolute security in mathematics, starting from nothing. I could probably do that with some cool drugs and a bit of religuous brainwashing. My preferred method is more mundane: teach you reading and writing and basic math, expose you to the wealth of human knowledge, train you in critical thinking, and then have you visit a course in mathematical logic. These are the true foundation of mathematics and logic.

If a logician is not allowed to use "and", "or" and other common words, then perhaps you should explain what he is allowed to use. Nothing? Surely, that is a bit too harsh. If you think about how one might achieve "perfect security" in mathematics, you will of course be reduced to questions about humans and the physical world, which lie outside the scope of mathematics and mathematical logic. All that logicians offer is a a mathematical analysis and reflection upon the nature of mathematics. And that is a very valuable thing to have.

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Do you plan to expand this? If not, you should, IMO, at least make this CW. –  quid Oct 25 '12 at 12:06
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quid, I think he posted this so he could write a proper answer even if the question gets closed. –  Asaf Karagila Oct 25 '12 at 12:10
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As I did in your answer to the similar question, I don't agree with this viewpoint. Of course when you intend Logic simply as a branch of mathematics, it is not different from number theory or differential geometry. But you can also intend the word "Logic" as what determines the inferences in formal theories, and formal theories are -so far- the only method we have to provide a rigorous foundation to mathematics. (...) –  Qfwfq Oct 25 '12 at 17:32
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To condense Andrej's remarks, Wittgenstein said something like: "Logic is the foundation of mathematics only as the painted rock is the support of the painted tower." –  Alexander Woo Oct 25 '12 at 20:05
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I find myself agreeing with Wittgenstein a bit more every year. –  Andrej Bauer Oct 25 '12 at 20:21
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